Sample Problem

Population growth: 1

a. A traditional problem in the mathematics of exponentials is the following:

"According to George Gamow, chess was invented by Sissa ben Dahir, Wazir of the court of King Shiram. King Shiram loved the game so much that he offered Sissa any reward he could name. Perhaps trying to impress the king with his mathematical skills, Sissa asked for some rice, one grain on the first square of the chessboard, two on the second, four on the third, eight on the fourth, and so on, each square's amount being the double of the previous square's. How much rice did Shiram owe Sissa?"*

Since the amount on the first square is 1, on the second is 2, the third is 2², etc., since there are 64 squares on the chessboard, the final square gets 2⁶³ grains (and the total is 2⁶⁴ -1). To get a sense for how much things grow after various doublings, evaluate the following in scientific notation:

amount after 5 doublings: 2⁵ =
amount after 10 doublings: 2¹⁰ =
amount after 20 doublings: 2²⁰ =
amount after 50 doublings: 2⁵⁰ =
amount after 64 doublings: 2⁶⁴ =

Estimate the mass of rice that would be on the chessboard.

b. An streptococcus bacterium can reproduce itself in about 30 minutes if there is adequate nutritional materials and appropriate conditions. If one bacterium gets in your system and you have no immune mechanism to destroy them or limit their growth, how many would there be in day if they all reproduced freely without restraint? Assuming they are using the materials in your body to build themselves, estimate what fraction of your body mass they would have consumed in one day. (You will need to find the approximate mass of a streptococcus bacterium.)

c. The population of the world is currently growing by about 1% per year. How many years will it take for the population of the world to double? (If you don't remember how to do exponentials, do it by hand.)

* Taken from http://classes.yale.edu/fractals/Chaos/Doubling/Doubling.html (but this is a standard problem).

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Page last modified February 6, 2008: G30